Multinomial Distribution
The multinomial distribution is an extension of the binomial distribution involving joint probabilities. It involves a similar statistical experiment, but this time there are more than two possible outcomes. Specifically, each trial can result in any of the k events E1 , E2 , ...., Ek , with respective probabilities p1 , p2 , .... , pk. In this case, the multinomial distribution is the joint probability distribution of the set of random variables X1 , X2 , ...., Xk , where Xi is the number of occurrences of Ei , i = 1, 2, ...., k, in n independent trials. It has a probability mass function of the following form:
The multinomial term 
epresents the number of ways distribute x1 outcomes of E1 , x2 outcomes of E2 , . . . xk outcomes of Ekn trials. among
The term 
is the probability that there are x1 outcomes of E1, x2 outcomes of E2 , xk outcomes of Ek.
The products of these two terms is the probability that in n trials, there are x1 outcomes for E1, x2 outcomes for E2,. . . xk outcomes for Ek.
EX. On average, Mark has a 50 % probability of not getting a hit during an at-bat opportunity. His probabilities are 12.5 % for a single, 10 % for a double, 2.5 % for a triple and 5 % for a home run. He gets a walk 20% of the time. The probability distribution for the number of each type of hit, as well as outs and walks, in n at-bats is modeled as follows:
Let random variable
| X1 | = number of outs | p1 = 0.500 |
| X2 | = number of singles | p2 = 0.125 |
| X3 | = number of doubles | p3 = 0.100 |
| X4 | = number of triples | p4 = 0.025 |
| X5 | = number of home runs | p5 = 0.050 |
| X6 | = number of walks | p6 = 0.200 |
The probability that Mark hits for the cycle (gets a single, double, triple and home run) in the next four at-bats is
p(0, 1, 1, 1, 1, 0; 4)
